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运筹学学报 >

2025 , Vol. 29 >Issue 2: 158 - 174

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2025.02.012

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一类线性反问题的变尺度外推硬阈值算法

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  • 山西师范大学数学与计算机科学学院, 山西太原 030031
张雪 E-mail: zhangxue2100442@163.com

收稿日期: 2022-04-27

  网络出版日期: 2025-06-12

基金资助

国家自然科学基金(11901368)

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运筹学学报编辑部, 2025, 版权所有，未经授权，不得转载。
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A variable metric extrapolation hard threshold algorithm for some linear inverse problem

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  • School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, Shanxi, China

Received date: 2022-04-27

  Online published: 2025-06-12

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, 2025, All rights reserved. Unauthorized reproduction is prohibited.
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摘要

稀疏正则化模型在信号和图像处理等反问题中有很广泛的应用。本文主要研究线性最小二乘$\ell_0$极小化问题的快速求解方法。外推向前向后分裂算法是最流行的求解方法之一。根据$\ell_0$正则化问题和该算法的特点, 我们将快速收敛的拟牛顿方法合理地应用于外推步中, 进而提出了一种分块变尺度外推算法, 并证明了其收敛性行为。我们在理论上证明了其快速性: 该方法具有线性收敛率, 甚至超线性收敛率。最后, 我们也通过数值实验展示了本文算法的有效性和快速性。

关键词： 分块; 变尺度; 外推; 线性收敛率; 超线性收敛率; ?0正则化

本文引用格式

张玉茹, 张雪, 兰茹 . 一类线性反问题的变尺度外推硬阈值算法[J]. 运筹学学报, 2025 , 29(2) : 158 -174 . DOI: 10.15960/j.cnki.issn.1007-6093.2025.02.012

Abstract

Sparsity regularization model is widely used in inverse problems such as signal and image processing. This paper mainly focuses on the linear least squares $\ell_0$ minimization problem coming from linear inverse problems. Extrapolation forward-backward splitting algorithm is one of the most popular solving methods. If the iteration number is sufficiently large, the non-zero index set of iteration point remains unchanged. Then extrapolation forward-backward splitting algorithm methods is equivalent to solving the problem $\min\limits_{x\in C} f(x)$ where $C$ is some linear subspace related to iteration points. On the other hand, variable metric type method can reach fast performance in practice. Encouraged by these, we employ the fast convergent quasi Newton method into the extrapolation step, and then propose a block variable metric extrapolation algorithm. Meanwhile, its convergence, linear convergence rate and superlinear convergence rate are studied. Finally, numerical experiments show the effectiveness and fast-speed of the proposed algorithm.

Key words： block; variable metric; extrapolation; linear convergence rate; super-linear convergence rate; ?0 regularization

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